78 XH TOPOLOGY AND TOPOLOGICAL ALGEBRA PROBLEMS 1. Let E be a vector space over K (= R or C) and let F be a vector subspace of KE whose elements are linear forms on E. We write <#, *'> in place of x'(x) for x e E and x' e F, Given a finite sequence (xDi *?*«$« of elements of F, show that for this sequence to be free (A. 4.1) it is necessary and sufficient that there should exist n vectors *i (1 Ss i ^ n) in E such that <xf , *J> = 8tJ (Kronecker delta). The vector space E is then the direct sum of the subspace V of dimension n generated by the xt, and the subspace W of all x e E such that <x, x'ty = 0 for 1 <; / g n. A linear form x' e F is such that <#, x'y = 0 for all x e W if and only if x' is a linear combination of the x\ . (Prove these assertions simultaneously by induction on n.) 2. The notation and hypotheses are the same as in Problem 1. Suppose also that there exists no vector x ^ 0 in E such that <x, #'> = 0 for allx' e F. (a) Show that the weak topology on F is metrizable if and only if E has an at most denumerable basis (A. 4.1). (Observe that if CvOi^^m and (Z/)I!S^B are two finite sequences of vectors in E such that the relation sup !<>>*, x'>l ^1 implies the relation sup |<X/,;O| ^ 1 in F, then the Zj are linear combinations of the yt. Show that if we have O>f , jc'> = Q for I ^i^m, then we must also have <z/ , #'> = 0 for 1 f£./ ^ n, and apply the result of Problem 1 by identifying E with the set of linear forms *'h-» (x, x'y on F.) (b) Deduce from (a) that, in an infinite-dimensional separable Hilbert space, the weak topology is not metrizable (cf. Section 5.9, Problem 2). (c) Show that every linear form on F which is continuous with respect to the weak topology can be written x'i* <x, x'y for a uniquely determined x e E (argue as in (a)). 3. (a) In a Hilbert space E, let A be a closed convex set (Section 12.14, Problem 11) and a a point of E. Show that there exists a unique point b e A such that d(a, A) = d(a, b) (the projection of a on A). For each z e A show that (z b\b a)*ZQ (argue as in (6.3.1)). (b) Deduce from (a) that every closed convex set A in E is the intersection of closed half-spaces whose frontiers are hyperplanes of support of A (Section 5.8, Problem 3). Deduce that A is weakly closed, and that if C is any convex subset of E, the strong and weak closures of C are the same. Every bounded closed convex subset of E is weakly compact. (c) Show that the weak closure in E of the sphere S : ||x|| = 1 is the ball ||x|| g 1. A subset of S is compact in the weak topology if and only if it is compact in the strong topology (use (12.15.12)). (d) Let U be a nonempty convex open set in E. Show that, if a is any frontier point of U, there exists a closed hyperplane of support of U passing through a (Hahn-Banach theorem for a Hilbert space). (Consider a sequence (an) of points exterior to U, con- verging to a, and for each n let bn be the projection of an on U defined in (a). Put unz=(bn a»)/\\bn an\\, and show that the sequence (un) has a subsequence which converges weakly to a point u ^ 0, by using the fact that U has an interior point c, and that this implies a relation of the form (c bn \ un) ^ r > 0 for all n.) (e) Suppose that E is separable, and let (en)nm be basis (6.5.2) of E. Let A be the closed convex hull (Section 12.14, Problem 13) of the set of points ±en/n. Show that A is compact and that the interior of A is empty, and that A has no closed hyperplane of support containing the (frontier) point 0, although there exist lines D passing through 0 and such that D n A « {0}.o I, which is contained in the Banach space 3fR(I) (7.2.1). Show that thevery set consisting of a single point is closed,